Shtukas and the Taylor expansion of $L$-functions

Abstract

We define the Heegner--Drinfeld cycle on the moduli stack of Drinfeld Shtukas of rank two with $r$-modifications for an even integer $r$. We prove an identity between (1) The $r$-th central derivative of the quadratic base change $L$-function associated to an everywhere unramified cuspidal automorphic representation $\pi$ of $PG{L}_2$; (2) The self-intersection number of the $\pi$-isotypic component of the Heegner--Drinfeld cycle. This identity can be viewed as a function-field analog of the Waldspurger and Gross--Zagier formula for higher derivatives of $L$-functions.

97 pages; minor revision

Cited by in corpus (2)

• Super-positivity of a family of L-functions
• The Gross-Zagier-Zhang formula over function fields